Underdoped, optimally doped, and overdoped superconductors ${\mathrm{TlSr}}_2$(${\mathit{R}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Ca}}_{\mathit{x}}$)${\mathrm{Cu}}_2$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ (R=Lu or Tl)

The crystal structures of underdoped, optimally doped, and overdoped ${\mathrm{TlSr}}_2$(${\mathit{R}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ca}}_{\mathit{x}}$)${\mathrm{Cu}}_2$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ (R=Lu or Tl) are refined from neutron powder-diffraction data. Carrier concentrations in this system are estimated from simple ionic consideration using the refined occupation numbers. The plot of ${\mathit{T}}_{\mathit{c}}$ vs carrier concentration shows the characteristic bell-shaped dependence. The in-plane bond length Cu-O(1) is affected by only the change of carrier concentration and provides a measurement of the doping level, while the apical bond length Cu-O(2) is affected by both the change of carrier concentration and the amount of cation substitution. If we take the effect of cation substitution into account, these bond lengths change ‘‘linearly’’ with carrier concentration over the entire range from the underdoped, through the optimally doped, to the overdoped region.